Authentic Task: Plan a Subdivi- sionrichard/MSP/talks/constr.pdf · Authentic Task: Plan a...

Authentic Task: Plan a Subdivi-sion

Professor Richard [email protected]

Dept. of Mathematical SciencesNorthern Illinois University

Website: http://math.niu.edu/∼richard

– p. 1

Construction ProblemA contractor builds two types of homes: the standardmodel and the deluxe model. The standard modelrequires one lot, $12,000 capital, 150 labor-days tobuild, and is sold for a profit of $2400. The deluxemodel requires one lot, $32,000 capital, 200labor-days to build, and is sold for a profit of $3400.The contractor has 150 lots. The bank is willing toloan him $2,880,000 for the project and he has amaximum labor force available of 24,000 labor-days.How many houses should he build to realize thegreatest profit?

– p. 2

OverviewWe will show how to solve “The ConstructionProblem”

OverviewWe will show how to solve “The ConstructionProblem”Good points:


• It is a nice example of an authentic task.



• It helps you review concepts about lines andintercepts.




Bad points:




Bad points:

• It is complicated.




Bad points:


• The numbers are unrealistic.




Bad points:


• The numbers are unrealistic.

• It is not a DIY problem. You will need guidanceto obtain a solution.

Some RemarksSome problems involve learning a precise method.

Some RemarksSome problems involve learning a precise method.Examples:


• Knitting


• Knitting

• Eating with chopsticks


• Knitting


Other problems are more suited to an independentapproach.


• Knitting



• Games, such as Tic-tac-toe or Sudoku


• Knitting



• Games, such as Tic-tac-toe or Sudoku

• The cell phone tower problem

Organizing the Data

Resource Standard Deluxe Available

Organizing the Data


Lot

Organizing the Data


Lot

Capital

Organizing the Data


Lot

Capital

Labor

Organizing the Data


Lot

Capital

Labor

Profit

Organizing the Data


Lot 1 1

Capital

Labor

Profit

Organizing the Data


Lot 1 1

Capital 12,000 32,000

Labor

Profit

Organizing the Data


Lot 1 1

Capital 12,000 32,000

Labor 150 200

Profit

Organizing the Data


Lot 1 1

Capital 12,000 32,000

Labor 150 200

Profit 2400 3400

Organizing the Data


Lot 1 1 150

Capital 12,000 32,000

Labor 150 200

Profit 2400 3400

Organizing the Data


Lot 1 1 150

Capital 12,000 32,000 2,880,000

Labor 150 200

Profit 2400 3400

Organizing the Data


Lot 1 1 150

Capital 12,000 32,000 2,880,000

Labor 150 200 24,000

Profit 2400 3400

Organizing the Data


Lot 1 1 150

Capital 12,000 32,000 2,880,000

Labor 150 200 24,000

Profit 2400 3400 P

Organizing the Data


Lot 1 1 150

Capital 12,000 32,000 2,880,000

Labor 150 200 24,000

Profit 2400 3400 P

Resource Inequalities:

Organizing the Data


Lot 1 1 150

Capital 12,000 32,000 2,880,000

Labor 150 200 24,000

Profit 2400 3400 P


• Lot 1x+ 1y ≤ 150

Organizing the Data


Lot 1 1 150

Capital 12,000 32,000 2,880,000

Labor 150 200 24,000

Profit 2400 3400 P


• Lot 1x+ 1y ≤ 150

• Capital 12000x+ 32000y ≤ 2880000

Organizing the Data


Lot 1 1 150

Capital 12,000 32,000 2,880,000

Labor 150 200 24,000

Profit 2400 3400 P


• Lot 1x+ 1y ≤ 150

• Capital 12000x+ 32000y ≤ 2880000

• Labor 150x+ 200y ≤ 24000

Organizing the Data


Lot 1 1 150

Capital 12,000 32,000 2,880,000

Labor 150 200 24,000

Profit 2400 3400 P


• Lot 1x+ 1y ≤ 150

• Capital 12000x+ 32000y ≤ 2880000

• Labor 150x+ 200y ≤ 24000

Profit Equation:

Organizing the Data


Lot 1 1 150

Capital 12,000 32,000 2,880,000

Labor 150 200 24,000

Profit 2400 3400 P


• Lot 1x+ 1y ≤ 150

• Capital 12000x+ 32000y ≤ 2880000

• Labor 150x+ 200y ≤ 24000

Profit Equation:

• Profit P = 2400x+ 3400y

Simplify the Math

• Resource Inequalities and Profit:

Simplify the Math


• Lot 1x+ 1y ≤ 150

Simplify the Math


• Lot 1x+ 1y ≤ 150• Capital 12000x+ 32000y ≤ 2880000

Simplify the Math


• Lot 1x+ 1y ≤ 150• Capital 12000x+ 32000y ≤ 2880000• Labor 150x+ 200y ≤ 24000

Simplify the Math


• Lot 1x+ 1y ≤ 150• Capital 12000x+ 32000y ≤ 2880000• Labor 150x+ 200y ≤ 24000• Profit P = 2400x+ 3400y

Simplify the Math



• Math Simplification

Simplify the Math



• Math Simplification• Divide Capital Inequality by 4000

Simplify the Math



• Math Simplification• Divide Capital Inequality by 4000• Divide Labor Inequality by 50

Simplify the Math




• Simplified Inequalities:

Simplify the Math




• Simplified Inequalities:• Lot x+ y ≤ 150

Simplify the Math




• Simplified Inequalities:• Lot x+ y ≤ 150• Capital 3x+ 8y ≤ 720

Simplify the Math




• Simplified Inequalities:• Lot x+ y ≤ 150• Capital 3x+ 8y ≤ 720• Labor 3x+ 4y ≤ 480

Simplify the Math




• Simplified Inequalities:• Lot x+ y ≤ 150• Capital 3x+ 8y ≤ 720• Labor 3x+ 4y ≤ 480• Profit P = 2400x+ 3400y

Graph the lot inequality

The graph of


The graph ofLot x+ y ≤ 150


The graph ofLot x+ y ≤ 150is the set of all points below the line



x+ y = 150



x+ y = 150

How do we draw this line?

Graphing the line

You may recall from the Algebra Class that you took

Graphing the line

You may recall from the Algebra Class that you tookin a previous lifetime

Graphing the line

You may recall from the Algebra Class that you tookthat graphing lines involves slopes and intercepts.

Graphing the line

You may recall from the Algebra Class that you tookthat graphing lines involves slopes and intercepts.In this case, the easiest method is to find the x− andy−intercepts and draw the line through them.

Graphing the line

You may recall from the Algebra Class that you tookthat graphing lines involves slopes and intercepts.In this case, the easiest method is to find the x− andy−intercepts and draw the line through them.The x−intercept is the point where the line crossesthe x−axis,

Graphing the line

You may recall from the Algebra Class that you tookthat graphing lines involves slopes and intercepts.In this case, the easiest method is to find the x− andy−intercepts and draw the line through them.The x−intercept is the point where the line crossesthe x−axis, or when y = 0.

Graphing the line

You may recall from the Algebra Class that you tookthat graphing lines involves slopes and intercepts.In this case, the easiest method is to find the x− andy−intercepts and draw the line through them.The x−intercept is the point where the line crossesthe x−axis, or when y = 0.Plugging y = 0 into the equation x+ y = 150, we see

Graphing the line

You may recall from the Algebra Class that you tookthat graphing lines involves slopes and intercepts.In this case, the easiest method is to find the x− andy−intercepts and draw the line through them.The x−intercept is the point where the line crossesthe x−axis, or when y = 0.Plugging y = 0 into the equation x+ y = 150, we seex = 150

Graphing the line

You may recall from the Algebra Class that you tookthat graphing lines involves slopes and intercepts.In this case, the easiest method is to find the x− andy−intercepts and draw the line through them.The x−intercept is the point where the line crossesthe x−axis, or when y = 0.Plugging y = 0 into the equation x+ y = 150, we seex = 150So the x−intercept is the point (150, 0).

Graphing the line

You may recall from the Algebra Class that you tookthat graphing lines involves slopes and intercepts.In this case, the easiest method is to find the x− andy−intercepts and draw the line through them.The x−intercept is the point where the line crossesthe x−axis, or when y = 0.Plugging y = 0 into the equation x+ y = 150, we seex = 150So the x−intercept is the point (150, 0).Similarly the y−intercept is the point (0, 150).

Graph x + y = 150

(0,0)✉

Graph x + y = 150

(0,0)✉

(150,0)

(0,150)

✉

✉

Graph x + y = 150

(0,0)✉

(150,0)

(0,150)

✉

✉

Lot x+ y = 150

DIY: Graph the capital line

Graph


GraphCapital 3x+ 8y = 720


GraphCapital 3x+ 8y = 720by finding the x− and y−intercepts.

Graph 3x + 8y = 720

(0,0)✉

Graph 3x + 8y = 720

(0,0)✉

(240,0)

(0,90)

✉

✉

Graph 3x + 8y = 720

(0,0)✉

(240,0)

(0,90)

✉

✉

Capital 3x+8y=720

Graph the labor line

Graph


GraphLabor 3x+ 4y = 480


GraphLabor 3x+ 4y = 480by finding the x− and y−intercepts.

Graph 3x + 4y = 480

(0,0)q

Graph 3x + 4y = 480

(0,0)q

Labor 3x+4y=480

(160,0)

(0,120)

q

q

Complete Graph

(0,0)q

Complete Graph

(0,0)q

Lot x+ y = 150

(150,0)

(0,150)

q

q

Complete Graph

(0,0)q

Lot x+ y = 150

(150,0)

(0,150)

q

q

Capital 3x+8y=720

(240,0)

(0,90)

q

q

Complete Graph

(0,0)q

Lot x+ y = 150

(150,0)

(0,150)

q

q

Capital 3x+8y=720

(240,0)

(0,90)

q

q

Labor 3x+4y=480

(160,0)

(0,120)

q

q

Solving Linear ProgrammingProblems

• The checkered region in this graph is called thefeasible region.



• It is the set of points satisfying the constraininginequalities.




• When the feasible set is finite, and the profitfunction is linear, the maximum (or minimum) isfound by the following





Linear Programming Rule:





Linear Programming Rule:

Maximums (and minimums) occur at corner points

Where are the corner points?

Looking at the complete graph, we see there are fivecorner points.


Looking at the complete graph, we see there are fivecorner points.Three of them are on the graph:


Looking at the complete graph, we see there are fivecorner points.Three of them are on the graph:(0,0)


Looking at the complete graph, we see there are fivecorner points.Three of them are on the graph:(0,0)(0,90)


Looking at the complete graph, we see there are fivecorner points.Three of them are on the graph:(0,0)(0,90)(150,0)


Looking at the complete graph, we see there are fivecorner points.Three of them are on the graph:(0,0)(0,90)(150,0)The other two corner points are


Looking at the complete graph, we see there are fivecorner points.Three of them are on the graph:(0,0)(0,90)(150,0)The other two corner points arethe intersection of the capital and labor lines and


Looking at the complete graph, we see there are fivecorner points.Three of them are on the graph:(0,0)(0,90)(150,0)The other two corner points arethe intersection of the capital and labor lines andthe intersection of the lot and labor lines.


Looking at the complete graph, we see there are fivecorner points.Three of them are on the graph:(0,0)(0,90)(150,0)The other two corner points arethe intersection of the capital and labor lines andthe intersection of the lot and labor lines.What about the intersection of the lot and capitallines?

DIY: Where do Capital and La-bor Lines Cross?Find the intersection of the lines:

DIY: Where do Capital and La-bor Lines Cross?Find the intersection of the lines:

Capital 3x+ 8y = 720

Labor 3x+ 4y = 480

Where do Capital and LaborLines Cross?

• Write the equations:


• Write the equations:Capital 3x+ 8y = 720

Labor 3x+ 4y = 480

Subtract 4y = 240



Labor 3x+ 4y = 480

Subtract 4y = 240

• Divide by 4:



Labor 3x+ 4y = 480

Subtract 4y = 240

• Divide by 4:

• So y =240

4= 60



Labor 3x+ 4y = 480

Subtract 4y = 240

• Divide by 4:

• So y =240

4= 60

• Plug y = 60 back into the Labor equation3x+ 4y = 480

Capital and Labor Line Inter-section

3x+ 4 · 60 = 480


3x+ 4 · 60 = 4803x+ 240 = 480


3x+ 4 · 60 = 4803x+ 240 = 4803x = 480− 240 = 240


3x+ 4 · 60 = 4803x+ 240 = 4803x = 480− 240 = 240x = 240

3= 80


3x+ 4 · 60 = 4803x+ 240 = 4803x = 480− 240 = 240x = 240

3= 80

• The intersection point is (80, 60).

DIY: Where do Labor and LotLines Cross?Find the intersection of the lines:

DIY: Where do Labor and LotLines Cross?Find the intersection of the lines:

Labor 3x+ 4y = 480

Lot x+ y = 150

Where do Labor and Lot LinesIntersect?

• Write the equations:


• Write the equations:Labor 3x+ 4y = 480

Lot x+ y = 150



Lot x+ y = 150

• Multiply Lot equation by 3



Lot x+ y = 150

• Multiply Lot equation by 3Capital 3x+ 4y = 480

Lot 3x+ 3y = 450

Subtract y = 30



Lot x+ y = 150


Lot 3x+ 3y = 450

Subtract y = 30

• Plug y = 30 back into the Lot equation



Lot x+ y = 150


Lot 3x+ 3y = 450

Subtract y = 30


• x+ 30 = 150 or x = 120



Lot x+ y = 150


Lot 3x+ 3y = 450

Subtract y = 30


• x+ 30 = 150 or x = 120

• The intersection point is (120, 30)

Graph

(0,0)t

Graph

(0,0)t

Lot x+ y = 150

(150,0)

(0,150)

t

q

Graph

(0,0)t

Lot x+ y = 150

(150,0)

(0,150)

t

q

Capital 3x+8y=720

(240,0)

(0,90)

q

t

Graph

(0,0)t

Lot x+ y = 150

(150,0)

(0,150)

t

q

Capital 3x+8y=720

(240,0)

(0,90)

q

t

Labor 3x+4y=480

(160,0)

(0,120)

q

q

Graph

(0,0)t

Lot x+ y = 150

(150,0)

(0,150)

t

q

Capital 3x+8y=720

(240,0)

(0,90)

q

t

Labor 3x+4y=480

(160,0)

(0,120)

q

q

(80,60)

(120,30)

t

tFeasible Set

Solving the Problem

x y P = 2400x+ 3400y

Solving the Problem

x y P = 2400x+ 3400y

0 0

Solving the Problem

x y P = 2400x+ 3400y

0 0

0 90

Solving the Problem

x y P = 2400x+ 3400y

0 0

0 90

80 60

Solving the Problem

x y P = 2400x+ 3400y

0 0

0 90

80 60

120 30

Solving the Problem

x y P = 2400x+ 3400y

0 0

0 90

80 60

120 30

150 0

Solving the Problem

x y P = 2400x+ 3400y

0 0 0

0 90

80 60

120 30

150 0

Solving the Problem

x y P = 2400x+ 3400y

0 0 0

0 90 $306,000

80 60

120 30

150 0

Solving the Problem

x y P = 2400x+ 3400y

0 0 0

0 90 $306,000

80 60 $396,000

120 30

150 0

Solving the Problem

x y P = 2400x+ 3400y

0 0 0

0 90 $306,000

80 60 $396,000

120 30 $390,000

150 0

Solving the Problem

x y P = 2400x+ 3400y

0 0 0

0 90 $306,000

80 60 $396,000

120 30 $390,000

150 0 $360,000

Solving the Problem

x y P = 2400x+ 3400y

0 0 0

0 90 $306,000

80 60 $396,000

120 30 $390,000

150 0 $360,000

Maximum Profit

Solving the Problem

x y P = 2400x+ 3400y

0 0 0

0 90 $306,000

80 60 $396,000

120 30 $390,000

150 0 $360,000

Maximum Profit

• 80 standard

Solving the Problem

x y P = 2400x+ 3400y

0 0 0

0 90 $306,000

80 60 $396,000

120 30 $390,000

150 0 $360,000

Maximum Profit

• 80 standard

• 60 deluxe

Solving the Problem

x y P = 2400x+ 3400y

0 0 0

0 90 $306,000

80 60 $396,000

120 30 $390,000

150 0 $360,000

Maximum Profit

• 80 standard

• 60 deluxe

• $396,000 profit

Solving the Problem

x y P = 2400x+ 3400y

0 0 0

0 90 $306,000

80 60 $396,000

120 30 $390,000

150 0 $360,000

Maximum Profit

• 80 standard

• 60 deluxe

• $396,000 profit

• Note: 10 lots are unused

Authentic Task: Plan a Subdivi- sionrichard/MSP/talks/constr.pdf · Authentic Task: Plan a...

Documents

Transcript of Authentic Task: Plan a Subdivi- sionrichard/MSP/talks/constr.pdf · Authentic Task: Plan a...